Fairview High School has an anime (Japanese animation) club that any student can attend. The relative frequency table shows the proportion of students in the high school who take Japanese and/or are in the anime club.

\begin{tabular}{|c|c|c|c|}
\hline & Take Japanese & \begin{tabular}{c}
Do not take \\
Japanese
\end{tabular} & Total \\
\hline In anime club & 0.15 & 0.01 & 0.16 \\
\hline Not in anime club & 0.05 & 0.79 & 0.84 \\
\hline Total & 0.20 & 0.80 & 1.0 \\
\hline
\end{tabular}

Given that a student takes Japanese, what is the likelihood that he or she is in the anime club?



Answer :

To determine the likelihood that a student is in the anime club given that they take Japanese, we will use the concept of conditional probability. Conditional probability is the probability of event [tex]\( A \)[/tex] occurring given that event [tex]\( B \)[/tex] has occurred, and it is calculated using the formula:

[tex]\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \][/tex]

Here, [tex]\( A \)[/tex] represents the event that a student is in the anime club and [tex]\( B \)[/tex] represents the event that a student takes Japanese. We need to find [tex]\( P(A|B) \)[/tex], which is the probability that a student is in the anime club given that they take Japanese.

From the table provided:
- [tex]\( P(A \cap B) \)[/tex] is the probability that a student is both in the anime club and takes Japanese. According to the table, this probability is 0.15.
- [tex]\( P(B) \)[/tex] is the probability that a student takes Japanese. According to the table, this probability is 0.20.

Using the conditional probability formula:

[tex]\[ P(\text{in anime club}|\text{takes Japanese}) = \frac{P(\text{in anime club and takes Japanese})}{P(\text{takes Japanese})} \][/tex]

Substituting the values from the table:

[tex]\[ P(\text{in anime club}|\text{takes Japanese}) = \frac{0.15}{0.20} \][/tex]

This quotient yields:

[tex]\[ P(\text{in anime club}|\text{takes Japanese}) = 0.75 \][/tex]

Thus, given that a student takes Japanese, the likelihood that he or she is in the anime club is [tex]\( 0.75 \)[/tex] or 75%.